Paper The following article is Open access

KIDs like cones

and

Published 8 November 2013 © 2013 IOP Publishing Ltd
, , Citation Piotr T Chruściel and Tim-Torben Paetz 2013 Class. Quantum Grav. 30 235036 DOI 10.1088/0264-9381/30/23/235036

0264-9381/30/23/235036

Abstract

We analyze vacuum Killing Initial Data on characteristic Cauchy surfaces. A general theorem on existence of Killing vectors in the domain of dependence is proved, and some special cases are analyzed in detail, including the case of bifurcate Killing horizons.

Export citation and abstract BibTeX RIS

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

1. Introduction

Killing Initial Data (KIDs) are defined as initial data on a Cauchy surface for a spacetime Killing vector field. Vacuum KIDs on spacelike hypersurfaces are well understood (see [1, 10] and references therein). In the spacelike case they play a significant role by providing an obstruction to gluing initial data sets [4, 6].

The question of KIDs on light-cones has been recently raised in [14]. The object of this note is to analyze this, as well as KIDs on characteristic surfaces intersecting transversally. It turns out that the situation in the light-cone case is considerably simpler than for the spacelike Cauchy problem, which explains our title.

For definiteness we assume the Einstein vacuum equations, in dimensions n + 1, n ⩾ 3, possibly with a cosmological constant,

Equation (1.1)

Similar results can be proved for Einstein equations with matter fields satisfying well-behaved evolution equations.

1.1. Light-cone

Consider the (future) light-cone CO issued from a point O in an (n + 1)-dimensional spacetime $({\mathscr M},g)$, n ⩾ 3; by this we mean the subset of ${\mathscr M}$ covered by future-directed null geodesics issued from O. (We expect that our results remain true for n = 2; this requires a more careful analysis of some of the equations arising, which we have not attempted to carry out.) Let (xμ) = (x0, xi) = (x0, r, xA) be a coordinate system such that x0 vanishes on CO. In the theorem that follows the initial data for the sought-for Killing vector field are provided by a spacetime vector field $\overline{Y}{}$ which is defined on CO only.1 We will need to differentiate $\overline{Y}{}$ in directions tangent to CO, for this we need a covariant derivative operator which involves only derivatives tangent to the characteristic hypersurfaces. In a coordinate system such that the hypersurface under consideration is given by the equation x0 = 0, for the first derivatives the usual spacetime covariant derivative $\nabla _i \overline{Y}{}^\mu$ applies. However, the tensor of second spacetime covariant derivatives involves the undefined fields $\nabla _ 0 \overline{Y}{}^\mu$. To avoid this we set, on the hypersurface {x0 = 0},2

Equation (1.2)

Equation (1.3)

with an obvious similar formula for ${\mathrm{D}}_{i} {\mathrm{D}}_j \overline{Y}{}^\mu$. When the restriction $\overline{\nabla _0 Y{}_ \mu }{}$ to the hypersurface {x0 = 0} of the x0-derivative is defined we have

Clearly, $ {\mathrm{D}}_{i} {\mathrm{D}}_j \overline{Y}{}_0$ coincides with $\nabla _i \nabla _j Y_0 |_{\lbrace x^0=0\rbrace }$ when $\overline{Y}{} ^\mu$ is the restriction to the hypersurface {x0 = 0} of a Killing vector field Yμ, as then ∇0Y0 = 0. This is of key importance for our equations below.

In the adapted null coordinates of [2] we have $\Gamma ^0_{1i}=0$ on {x0 = 0} (see [2, appendix A]), so in these coordinates DiDj differs from ∇ij only when i, j ∈ {2, ..., n}.

Our main result is:

Theorem 1.1. Let $\overline{Y}{}$ be a continuous vector field defined along CO in a vacuum spacetime $({\mathscr M},g)$, smooth on CO∖{O}. There exists a smooth vector field X satisfying the Killing equations on D+(CO) and coinciding with $\overline{Y}{}$ on CO if and only if on CO it holds that

Equation (1.4)

Equation (1.5)

Furthermore, (1.5) is not needed on the closure of the set on which the divergence τ of CO is non-zero.

While this is not necessary, the analysis of KIDs on light-cones thus can be split into two cases: the first is concerned with the region sufficiently close to the tip of the cone where the expansion τ has no zeros. Once a spacetime with Killing field has been constructed near the vertex, the initial value problem for the remaining part of the cone can be reduced to a characteristic initial value problem with two transversally intersecting null hypersurfaces, which will be addressed in theorem 1.2. From this point of view the key restriction for light-cones is (1.4).

The proof of theorem 1.1 can be found in section 2.5. In order to prove it we will first establish some intermediate results, theorems 2.1 and 2.5 below, which require further hypotheses. It is somewhat surprising that these additional conditions turn out to be automatically satisfied.

Equations (1.4) provide thus necessary-and-sufficient conditions for the existence, to the nearby future of O, of a Killing vector field. They can be viewed as the light-cone equivalent of the spacelike KID equations, keeping in mind that (1.5) should be added when CO contains open subsets on which τ vanishes. As made clear by the definitions, (1.4) and (1.5) involve only the derivatives of $\overline{Y}{}$ in directions tangent to CO.

We shall see in section 2.6 that some of the equations (1.4) can be integrated to determine $\overline{Y}{}$ in terms of data at O. Once this has been done, we are left with the trace-free part of $ {{\mathrm{D}}_A \overline{Y}{}_B + {\mathrm{D}}_B \overline{Y}{}_A}=0$ as the 'reduced KID equations'.

It should be kept in mind that a Killing vector field satisfies an overdetermined system of second-order ODEs which can be integrated along geodesics starting from O, see (2.56) below. This provides both X, and its restriction $\overline{X}{}$ to CO, in a neighborhood of O, given the free data Xα|O and ∇Xβ]|O. We will see in the course of the proof of theorem 2.1 how such a scheme ties-in with the statement of the theorem, cf in particular section 2.3.

In section 2.6 we give a more explicit form of the KID equations (1.4) on a cone and discuss some special cases.

In section 3.3 we show how bifurcate Killing horizons arise from totally geodesic null surfaces normal to a spacelike submanifold S of co-dimension two, and how isometries of S propagate to the spacetime.

1.2. Two intersecting null hypersurfaces

Throughout, we employ the symbol ($ \breve{\cdot }$) to denote the trace-free part of the field (·) with respect to $\tilde{g} = \overline{g}{}_{AB}\, \mathrm{d}x^A\mathrm{d}x^B$. Further, an overbar denotes restriction to the initial surface.

In what follows the coordinates xA are assumed to be constant on the generators of the null hypersurfaces. The analogue of theorem 1.1 for two intersecting hypersurfaces reads:

Theorem 1.2. Consider two smooth null hypersurfaces N1 = {x1 = 0} and N2 = {x2 = 0} in an (n + 1)-dimensional vacuum spacetime $({\mathscr M},g)$, with transverse intersection along a smooth (n − 1)-dimensional submanifold S. Let $\overline{Y}{}$ be a continuous vector field defined on N1N2 such that $\overline{Y}{}|_{N_1}$ and $\overline{Y}{}|_{N_2}$ are smooth. There exists a smooth vector field X satisfying the Killing equations on D+(N1N2) and coinciding with $\overline{Y}{}$ on N1N2 if and only if on N1 it holds that

Equation (1.6)

Equation (1.7)

Equation (1.8)

Equation (1.9)

where D is the analogue on N1 of the derivative operator (1.2) and (1.3), with identical corresponding conditions on N2, and on S one needs further to assume that

Equation (1.10)

Equation (1.11)

Equation (1.12)

Equation (1.13)

Similarly to theorem 1.1, (1.9) can be replaced by the requirement that $\overline{g}{}^{AB} {\mathrm{D}}_A \overline{Y}{}_B=0$ on regions where the divergence of N1 is non-zero. An identical statement applies to N2.

Theorem 1.2 is proved in section 3. As before, (1.6)–(1.13) provide necessary-and-sufficient conditions for the existence, to the future of S, of a Killing vector field. Hence they provide the equivalent of the spacelike KID equations in the current setting. Note that in (1.6)–(1.13) the derivative D coincides with ∇.

2. The light-cone case

2.1. Adapted null coordinates

We use local coordinates (x0u, x1r, xA) adapted to the light-cone as in [2], in the sense that the cone is given by CO = {x0 = 0} . Further, the coordinate x1 parameterizes the null geodesics emanating to the future from the vertex of the cone, while the xA's are local coordinates on the level sets {x0 = 0, x1 = const}≅Sn − 1, and are constant along the generators. Then the metric takes the following form on CO:

Equation (2.1)

We stress that we do not assume that this form of the metric is preserved under differentiation in the x0–direction, i.e. we do not impose any gauge condition off the cone. On CO the inverse metric reads

Equation (2.2)

with

Equation (2.3)

2.2. A weaker result

We start with a weaker version of theorem 2.5 which, moreover, assumes that the vector field $\overline{Y}{}$ there is the restriction to the light-cone of some smooth vector field Y:

Theorem 2.1. Let Y be a smooth vector field defined in a neighborhood of CO in a vacuum spacetime $({\mathscr M},g)$. There exists a smooth vector field X satisfying the Killing equations on D+(CO) and coinciding with Y on CO if and only if the equations

Equation (2.4)

Equation (2.5)

Equation (2.6)

Equation (2.7)

are satisfied by the restriction $\overline{Y}{}$ of Y to CO.

Proof. To prove necessity, let X be a smooth vector field satisfying the Killing equations on D+(CO):

Equation (2.8)

the tangent components of which give (2.4). It further easily follows from (2.8) that X satisfies

Equation (2.9)

and (2.5)–(2.7) similarly follow; for (2.7) the equation $\overline{A}{}_{00}=0$ is used.

To prove sufficiency, by contracting (2.9) one finds

Equation (2.10)

(which equals −λXσ under (1.1)). So, should a solution X of our problem exist, it will necessarily satisfy the wave equation (2.10).

Now, it follows from e.g. [7, théorème 2] that for any smooth vector field Y defined on ${\mathscr M}$ there exists a smooth vector field X on ${\mathscr M}$ solving (2.10) to the future of O, such that,

Equation (2.11)

Here, and elsewhere, overlining denotes restriction to CO. Further $X|_{D^+(C_O)}$ is uniquely defined by $Y|_{C_O}$.

Applying $\Box$ to (2.8) leads to the identity

Equation (2.12)

When Xμ solves (2.10) this can be rewritten as a homogeneous linear wave equation for the tensor field Aμν,

Equation (2.13)

if one notes that, under (1.1) the last term $-2{\mathscr L}_X R_{\mu \nu }$ equals −2λAμν (and, in fact, cancels with the before-last one, though this cancellation is irrelevant for the current discussion). It follows from uniqueness of solutions of (2.13) that a solution X of (2.10) will satisfy the Killing equation on D+(CO) if and only if

Equation (2.14)

But by (2.4) we already have

Equation (2.15)

so it remains to show that the equations $\overline{A}{_{0\mu }}=0$ hold. (Annoyingly, these equations involve the derivatives $\overline{\partial _0 X_\mu }{}$ which cannot be expressed as local expressions involving only the initial data $\overline{X}{} = \overline{Y}{}$.) The theorem follows now directly from lemma 2.4 below. □

Definition 2.2. It is convenient to introduce, for a given vector field X, the tensor field

Equation (2.16)

Whenever it is clear from the context which vector field is meant we will suppress its appearance and simply write Sμνσ.

Using the algebraic symmetries of the Riemann tensor we find:

Lemma 2.3. It holds that:

  • (i)  
    2Sαβγ = 2∇Aβ)γ − ∇γAαβ,
  • (ii)  
    2Sα(βγ) = ∇αAβγ,
  • (iii)  
    S[αβ]γ = 0.

Lemma 2.4. Suppose that $\overline{A}{}_{ij}=0$ and $\Box X= -\lambda X$.

  • (1)  
    (2.5) is equivalent to $ \overline{A}{}_{01}=0 .$
  • (2)  
    If (2.5) holds, then (2.6) is equivalent to $ \overline{A}{}_{0A}=0 .$
  • (3)  
    If (2.5) and (2.6) hold, then (2.7) is equivalent to $ \overline{A}{}_{00}=0 .$

Proof. It turns out to be convenient to consider the identity

Thus, it holds that

Equation (2.17)

In adapted null coordinates (2.17) implies

Equation (2.18)

Due to lemma 2.3 we have

Equation (2.19)

When (ij) = (11) that yields

Equation (2.20)

Inserting into (2.18) with μ = 1, after some simplifications one obtains

Using the vanishing of the $\overline{\Gamma }{}^0_{i1}$'s [2, appendix A] and the $\overline{A}{}_{ij}$'s, this becomes a linear homogeneous ODE for $\overline{A}{}_{01}$; in the notation of the last reference (where, in particular, τ denotes the divergence of CO):3

Equation (2.21)

If $\overline{A}{}_{01}=0$, the vanishing of $\overline{S}{}_{110}$ immediately follows.

To prove the reverse implication, for definiteness we assume here and in what follows a coordinate system as in [2, section 4.5]. In this coordinate system τ behaves as (n − 1)/r for small r, ν0 satisfies ν0 = 1 + O(r2), and (2.21) is a Fuchsian ODE with the property that every solution which is o(r−(n − 1)/2) for small r is identically zero, see appendix A. As $\overline{A}{}_{01}$ is bounded, when $\overline{S}{}_{110}$ vanishes we conclude that

Equation (2.22)

This proves point 1 of the lemma.

Next, (2.18) with μ = D reads

Equation (2.23)

Using (2.19) with (ij) = (A1),

Equation (2.24)

to eliminate $ \overline{\nabla _0 A_{1 D}}{}$ from (2.23), and invoking (2.22), on CO one obtains a system of Fuchsian radial ODEs for $\overline{A}{}_{0D}$,

Equation (2.25)

with zero being the unique solution with the required behavior at r = 0 when $\overline{S}{}_{B10}=0$:

Equation (2.26)

This proves point 2 of the lemma.

Let us finally turn attention to (2.18) with μ = 0:

Equation (2.27)

The transverse derivatives $ \overline{\nabla _0 A _{1i}}{}$ can be eliminated using (2.20) and (2.24),

Equation (2.28)

The remaining one, $ \overline{g}{}^{AB}\overline{\nabla _0 A _{AB}}{}$, fulfils the following equation on CO, which follows from (2.19),

where we have set

Equation (2.29)

Note that $\tilde{S}$ is the negative of the left-hand side of (2.7), and we want to show that the vanishing of $\tilde{S}$ is equivalent to that of $\overline{A}{}_{00}$. Equation (2.28) with $\overline{A}_{ij}=0$ and $\overline{A}_{0i}=0$ (i.e. $\overline{S}{}_{i10}=0$) yields

Equation (2.30)

For $\tilde{S}=0$ this is again a Fuchsian radial ODE for $ \overline{A}{}_{0 0}$, with the only regular solution $ \overline{A}{}_{00}=0$, and the lemma is proved. □

2.3. The free data for X

Let us explore the nature of (1.4). Making extensive use of [2, appendix A], and of the notation there (thus $\kappa \equiv \overline{\Gamma }{}^1_{11}$, $\xi _A\equiv -2\overline{\Gamma }{}^{1}_{1A}$, while $\chi _{A}{}^B=\overline{\Gamma }{}_{1A}^B$ denotes the null second fundamental form of CO), we find

Equation (2.31)

Equation (2.32)

Equation (2.33)

For definiteness, in the discussion that follows we continue to assume a coordinate system as in [2, section 4.5], in particular κ = 0 and

Equation (2.34)

Equation (2.35)

Equation (2.36)

Under (1.4) the left-hand sides of (2.31)–(2.33) vanish. Hence, we can determine $\overline{X}{}_1$ by integrating (2.31),

Equation (2.37)

for some function of the angles.

We continue by integrating (2.32). This is a Fuchsian ODE for $\overline{X}{}_B$, the solutions of which are of the form

Equation (2.38)

where $\mathring{{\mathscr D}}$ is the covariant derivative operator of the unit round metric s on Sn − 1 where fA(xB) is an integration function.

In a neighborhood of O, where τ does not vanish, the component $\overline{X}{} ^1$ can be algebraically determined from the equation $\overline{A}_A{}^A=0$, leading to

Equation (2.39)

where Δs is the Laplace operator of the metric s.

The equation $\breve{\overline{A}}_{AB}=0$, where $\breve{\overline{A}}_{AB}$ denotes the $\tilde{g}$-trace free part of $ {\overline{A}}_{AB}$, imposes the relations

Equation (2.40)

Equation (2.41)

with ($ \breve{\cdot }$) denoting here the trace-free part with respect to the metric s.

We wish, now, to relate the values of c and fA to the values of the vector field $\overline{X}{}$ at the vertex, under the supplementary assumption that $\overline{X}{}$ is the restriction to CO of a differentiable vector field defined in spacetime. Following [2], we denote by yμ normal coordinates centered at O. Given the coordinates yμ the coordinates xα can be obtained by setting

Equation (2.42)

for some functions μA, so that the xA form local coordinates on Sn − 1, and

Equation (2.43)

We underline the components of tensor fields in the yα-coordinates, in particular

Equation (2.44)

For vector fields such that $\underline{X}{}^\mu$ is continuous, we obtain

Equation (2.45)

Thus, for such vector fields, $\overline{X}{}{}_1(0)$ is a linear combination of ℓ = 0 and ℓ = 1 spherical harmonics, and contains the whole information about $\underline{X}{}^\alpha (0)$. We conclude that

Equation (2.46)

Equation (2.40) will be satisfied if and only if c is of the form (2.46), which can be seen by noting that (2.46) provides a family of solutions of (2.40) with the maximal possible dimension.

To determine fA when $\underline{X}{}^\mu$ is differentiable at the origin we Taylor expand $\underline{X}{}$ there,

so that

Equation (2.47)

which determines fA in terms of $\underline{\partial _\mu X_i}{} (0)$. Equation (2.41), which is the conformal Killing vector field equation on Sn − 1, will be satisfied under the hypotheses of theorem 2.1 if and only if $\underline{\partial _i X_j}{}(0)$ is anti-symmetric.

2.4. A second intermediate result

As a next step toward the proof of theorem 1.1, we drop in theorem 2.1 the assumption of $\overline{Y}{}$ being the restriction of a smooth spacetime vector field:

Theorem 2.5. Let $\overline{Y}{}$ be a vector field defined along CO in a vacuum spacetime $({\mathscr M},g)$. There exists a smooth vector field X satisfying the Killing equations on D+(CO) and coinciding with $\overline{Y}{}$ on CO if and only if on CO it holds that

Equation (2.48)

Equation (2.49)

Equation (2.50)

Equation (2.51)

Proof of theorem 2.5. We wish to apply theorem 2.1. The crucial step is to construct the vector field Y needed there. For further reference we note that (2.51) will not be needed for this construction.

In the argument that follows we shall ignore the distinction between $\overline{X}{}$ and $\overline{Y}{}$ whenever it does not matter.

By hypothesis it holds that

Equation (2.52)

Equation (2.53)

We define an anti-symmetric tensor $\overline{F}{}_{\mu \nu }$ via

Then

Moreover,

With (2.52) that gives

Further,

To sum it up, (2.52) and (2.53) imply that the equations

Equation (2.54)

Equation (2.55)

hold on CO,

Let $\mathring{X}^\mu = \overline{X}^\mu |_ O$ and $\mathring{F}_{\mu \nu }= \overline{F}{}_{\mu \nu }|_ O$ be the initial data at O needed for solving those equations. These data can be calculated as follows: (2.40) and (2.41) show that $\mathring{{\mathscr D}}_A c$ and fA are conformal Killing fields on the standard sphere (Sn − 1, s). It follows from [13, proposition 3.2] (a detailed exposition can be found in [12, proposition 2.5.1]) that c is a linear combination of the first two spherical harmonics, so that $\mathring{X}^\mu$ can be read off from c using (2.46). Similarly (2.47) can be used to read off $\mathring{F}_{\mu \nu }$ from fA.

We conclude that, in coordinates adapted to CO as in (2.42), under the hypotheses of theorem 2.5 the desired Killing vector X is a solution of the following problem:

Equation (2.56)

Note that the first four equations above determine uniquely the initial data $\overline{X}{^\mu }$ on CO needed to obtain a unique solution of the wave equation for Xμ.

Now, we claim that there exists a smooth vector field Yμ defined near O so that $\overline{X}^\mu$ is the restriction of Yμ to the light-cone. To see this, let $\mathring{\ell }^\mu$ be given and define (xμ(s), Zμ(s), Fαβ(s)) as the unique solution of the problem

Equation (2.57)

For initial values such that xμ(1) is defined, set

Equation (2.58)

It follows from smooth dependence of solutions of ODEs upon initial data that Yμ is smooth in all initial variables, in particular in $\mathring{\ell }^\mu$. If the xμ's are normal coordinates centered at O, then $x^\mu (s=1)=\mathring{\ell }^\mu$, which implies that (2.58) defines a smooth vector field in a neighborhood of O. It then easily follows that the restriction of Yμ to CO equals $\overline{X}{}^\mu$, as defined by the first four equations in (2.56).

The hypotheses of theorem 2.1 are now satisfied, and theorem 2.5 is proved. □

2.5. Proof of theorem 1.1

To prove theorem 1.1 we will use theorem 2.1, together with some of the ideas of the proof of theorem 2.5. We need to show that (1.4) together with the Einstein equations imply both the existence of a smooth extension Y of $\overline{Y}{}$, and that (2.5)–(2.7) hold.

2.5.1. Properties of Sμνσ

Recall the definition

Equation (2.59)

and lemma 2.3.

In the context of theorem 1.1, only those components of the tensor field Sαβγ which do not involve ∂0-derivatives of X are apriori known. One easily checks:

Lemma 2.6. The components

Equation (2.60)

of the restriction to CO of Sμνσ can be algebraically $\overline{X}_\sigma\equiv \overline{Y}_\sigma, D_i \overline{X}_\sigma\equiv D_i \overline{Y}_\sigma \hbox{ and }D_i D_j \overline{X}_\sigma \equiv D_i D_j \overline{Y}_\sigma$.

We wish, next, to calculate ∇αSαβγ and ∇γSαβγ. This requires the knowledge of ∇0Xμ, of ∇00Xμ, and even of ∇000Xμ in some equations. For this, let X be any extension of $\overline{X}{}$ from the light-cone to a punctured neighborhood ${\mathscr O}\setminus \lbrace O\rbrace$ of O, so that the transverse derivatives appearing in the following equations are defined. X is assumed to be smooth on its domain of definition, and we emphasize that we do not make any hypotheses on the behavior of the extension X as the tip O of the light-cone is approached. As will be seen, the transverse derivatives of X on CO drop out from those final formulae which are relevant for us.

We will make use several times of

which is a standard consequence of the second Bianchi identity when the Ricci tensor is proportional to the metric.

We start with ∇αSαβγ. Two commutations of derivatives allow us to rewrite the first term in the divergence of Sαβγ over the first index as

Equation (2.61)

Hence, since Rαβ = λgαβ, and using the first Bianchi identity in the second line

Equation (2.62)

Similarly,

Equation (2.63)

Now, on CO and in coordinates adapted to the cone

Equation (2.64)

while

Equation (2.65)

In order to handle undesirable terms such as ∇0Sβγ1 we write

Equation (2.66)

2.5.2. Analysis of condition (2.5)

Lemma 2.7. Assume that $\overline{A}{}_{1i}=0$ and $\breve{\overline{A}{}}_{AB}=0$. Then, in vacuum,

Equation (2.67)

Proof. By lemma 2.3 the vanishing of $\overline{A}{}_{1i}$ implies

  • (i)  
    $\overline{S}{}_{111} =0$,
  • (ii)  
    $0=\overline{S}{}_{11A}$, as well as all permutations thereof.

Consider (2.63) with (βγ) = (11). Setting $a:= \overline{g}{}^{AB}\overline{A}{}_{AB}$ we find

Equation (2.68)

Due to lemma 2.3 we have

Using (2.65) with (βγ) = (11), as well as the last equation, and employing again lemma 2.3 we obtain, on CO,

Equation (2.69)

Using (2.66) we find

Equation (2.70)

Equating (2.68) with (2.69), and using the last equation we end up with (2.67). □

Corollary 2.8. In a region where the divergence τ does not vanish (in particular, near the vertex), $\overline{A}{}_{ij}=0$ implies, in vacuum, $\overline{S}{}_{110}=0$.

2.5.3. Analysis of condition (2.6)

Lemma 2.9. Assume that $\overline{A}{}_{ij}=0$ and $\overline{S}{}_{110}=0$. Then, in vacuum, $\overline{S}{}_{A10}=0$.

Proof. From (2.66) we obtain

This allows us to rewrite (2.65) with (βγ) = (A1) on CO as

Equation (2.71)

Combining with (2.63), which reads with (βγ) = (A1)

we obtain on the initial surface

Equation (2.72)

Assuming $\overline{A}{}_{ij}=0$, lemma 2.3 shows that $\overline{S}{}_{i11}=0= \overline{S}{}_{1i1}=\overline{S}{}_{11i}= \overline{S}{}_{A1B}$. This allows us to rewrite the right-hand side of (2.72) as

Now, using in addition that $\overline{S}{}_{110}=0$,

Hence,

and thus, again due to $\overline{S}{}_{110}=0$ and lemma 2.3,

Equation (2.73)

But zero is the only solution of this equation which is o(r−(n − 1)), and to be able to conclude that

Equation (2.74)

we need to check the behavior of $\overline{S}{}_{A10}$ at the vertex. For definiteness we assume a coordinate system as in [2, section 4.5]. Now, by definition,

From (2.37)–(2.39) we find

Equation (2.75)

Equation (2.76)

Using the formulae from [2, appendix A] one obtains

which implies that (2.74) holds, and lemma 2.9 is proved. □

2.5.4. Proof of theorem 1.1

We are ready now to prove our main result.

Proof of theorem 1.1. By assumption, using obvious notation, $\overline{A}{}^{(\overline{Y})}_{ij}=0$. When τ does not vanish corollary 2.8 applies and shows that $\overline{S}{}^{(\overline{Y})}_{110}=0$. Otherwise, $\overline{S}{}^{(\overline{Y})}_{110}=0$ holds by hypothesis and lemma 2.9 shows that $\overline{S}{}^{(\overline{Y})}_{A10}$ vanishes as well. In the proof of theorem 2.5 we have shown that $\overline{A}{}^{(\overline{Y})}_{ij}=0$ and $\overline{S}{}^{(\overline{Y})}_{i10}=0$ suffice to make sure that $\overline{Y}{}$ is the restriction to CO of a smooth spacetime vector field Y. Then, due to the Cagnac–Dossa theorem [7, théorème 2], there exists a smooth vector field X with $\overline{X}{}=\overline{Y}{}$ which solves $\Box X=-\lambda X$. The assertions of theorem 1.1 follow now from theorem 2.1, whose remaining hypotheses are satisfied by lemma 2.11 below. □

2.5.5. Analysis of condition (2.7)

A straightforward application of lemma 2.3 yields

Lemma 2.10. Assume that $\overline{A}{}_{i \mu }=0$. Then

  • (i)  
    $\overline{S}{}_{ijk}=0$,
  • (ii)  
    $\overline{S}{}_{110}=\overline{S}{}_{101} = \overline{S}{}_{011} = 0$,
  • (iii)  
    $\overline{S}{}_{A10} =\overline{S}{}_{A01}= \overline{S}{}_{1A0} = \overline{S}{}_{0A1} = \overline{S}{}_{10A} = \overline{S}{}_{01A} = 0$.

Lemma 2.11. Consider a smooth vector field X in a vacuum spacetime $({\mathscr M},g)$ which satisfies $\overline{A}{}_{i \mu }=0$ on CO and $\Box X + \lambda X=0$. Then

Equation (2.77)

Proof. Equation (2.62) yields with $\overline{A}{}_{i \mu }=0$, $\Box X + \lambda X=0$ and in vacuum

On the other hand, (2.64) gives with lemma 2.3 and 2.10 on CO,

Moreover, from (2.66) and lemma 2.3 we deduce that, on CO,

Hence,

Using the vacuum constraint [2] $0 = \lambda \overline{g}{}_{11} = \overline{R}{}_{11} = -\partial _r \tau + \kappa \tau -|\chi |^2$, we obtain

We employ (2.30),

which holds since all the hypotheses of lemma 2.4 are fulfilled, to end up with

Regularity at O in coordinates as in [2, section 4.5] gives $\tilde{S} = O(r^{-1})$, which implies that $\tilde{S}=0$ is the only possibility. □

2.6. Analysis of the KID equations in some special cases

2.6.1. KID equations

Theorem 1.1 shows that a vacuum spacetime emerging as solution of the characteristic initial value problem with data on a light-cone possesses a Killing field if and only if the conformal class $\gamma _{AB}=[\overline{g}{}_{AB}]$ of gAB, which together with κ describes the free data on the light-cone, is such that, in the region where τ has no zeros, the KIDequations $\overline{A}{}_{ij}=0$ admit a non-trivial solution $\overline{Y}{}$. Written as equations for the vector field $\overline{Y}{}$, they read (we use the formulae from [2, appendix A])

Equation (2.78)

Equation (2.79)

Equation (2.80)

Equation (2.81)

where σAB denotes the trace-free part of χAB, $\xi ^A:=\overline{g}{}^{AB}\xi _B$ and

Equation (2.82)

Equation (2.83)

The analysis of these equations is identical to that of their covariant counterpart, already discussed in section 2.3. The first three equations, arising from $\overline{A}{}_{1i}=0$ and $\overline{g}{}^{AB} \overline{A}{}_{AB}=0$ determine a class of candidate fields (depending on the integration functions c(xA) and fA(xB), with $\mathring{{\mathscr D}}_A c$ and fA being conformal Killing fields on (Sn − 1, s). Note that it is crucial for the expansion τ to be non-vanishing in order for $\overline{g}{}^{AB} \overline{A}{}_{AB}=0$ to provide an algebraic equation for $\overline{Y}{}^1$. Regardless of whether τ has zeros or not, we can determine $\overline{Y}{}^1$ by integrating radially (1.5), compare remark 3.2 below.

2.6.2. Killing vector fields tangent to spheres

Let us consider the special case where the spacetime admits a Killing field X with the property that $\overline{X}{}^0 = \overline{X}{}^1 =0$ on CO. The KID equations for the candidate field $\overline{Y}{}$ (2.78)–(2.81) then reduce to

which leads us to the following corollary of theorem 1.1.

Corollary 2.12. Consider initial data $ \bar{g}_{AB}(r,x^C)\, {\rm d}x^A\, {\rm d}x^B$ for the vacuum Einstein equations (cf, e.g., [5]) on a light-cone CO. In the resulting vacuum spacetime there exists a Killing field X with $\overline{X}{}^0 = \overline{X}{}^1 =0$ on CO defined on a neighborhood of the vertex O if and only if the family of Riemannian manifolds

admits an r-independent Killing field fA = fA(xB).

2.6.3. Killing vector fields tangent to the light-cone

Let us now restrict attention to those Killing fields which are tangent to the cone CO, i.e. we assume

Equation (2.84)

We start by noting that in the coordinates of (2.42) we have

for an anti-symmetric matrix ωμν. Hence, quite generally,

Equation (2.85)

Equation (2.86)

Thus (2.84) does not impose any restrictions on ωμν, and we have

Equation (2.87)

Next, under (2.84) the KID equations (2.78)–(2.81) for the candidate field $\overline{Y}{}$ become

Equation (2.88)

Equation (2.89)

Equation (2.90)

or, equivalently (note that $\partial _r \tilde{\Gamma }^B_{AB}= \partial _A\tau$)

Equation (2.91)

Equation (2.92)

Equation (2.93)

where we have set $f_A := \overline{g}{}_{AB}f^B$. Equations (2.91)–(2.93) provide thus a relatively simple form of the necessary-and-sufficient conditions for existence of Killing vectors tangent to CO.

If we choose a gauge where τ = (n − 1)/r (cf e.g. [5]), the last three equations become

Equation (2.94)

Equation (2.95)

Equation (2.96)

Note that there are no non-trivial Killing vectors tangent to all generators of the cone, $\overline{Y}{}^A=0$, as (2.95) gives then $\overline{Y}{}^1=0$. This should be contrasted with a similar question for intersecting null hypersurfaces, see section 3.3.

3. Two intersecting null hypersurfaces

3.1. An intermediate result

In analogy with the light-cone case let us first prove an intermediate result.

Theorem 3.1. Consider two smooth null hypersurfaces N1 = {x1 = 0} and N2 = {x2 = 0} in an (n + 1)-dimensional vacuum spacetime $({\mathscr M},g)$, with transverse intersection along a smooth submanifold S. Let $\overline{Y}{}$ be a vector field defined on N1N2. There exists a smooth vector field X satisfying the Killing equations on D+(N1N2) and coinciding with $\overline{Y}{}$ on N1N2 if and only if on N1 it holds that

Equation (3.1)

Equation (3.2)

Equation (3.3)

Equation (3.4)

Equation (3.5)

Equation (3.6)

where D is the analogue on N1 of the derivative operator (1.2)–(1.3); similarly on N2; while on S one needs further to assume that

Equation (3.7)

Proof. The proof is essentially identical to the proof of theorem 2.1. The candidate field is constructed as a solution of the wave equation (2.10); the delicate question of regularity of $\overline{Y}{}$ needed at the vertex in the cone case does not arise. Existence of the solution in J+(N1N2) follows from [11].

The main difference is that one cannot invoke regularity at the vertex to deduce the vanishing of, say on N2, $A_{2\mu }|_{N_2}$ from the equations which correspond to (2.21), (2.25) and (2.30). Instead, one needs further to require (3.7) as well as

However, the last two conditions follow from (3.1) and (3.2) on N1. □

3.2. Proof of theorem 1.2

We prove now our main result for transversally intersecting null hypersurfaces.

Proof of theorem 1.2. We want to show that (1.6)–(1.13) imply that all the remaining assumptions of theorem 3.1, namely (3.1)–(3.7), are satisfied. The conditions (3.1), (3.2), (3.4) and (3.7) follow trivially.

Lemma 2.7, adapted to the intersecting null hypersurfaces-setting, tells us that gAB(AYB) vanishes on N1N2 due to (1.11) and (1.12). Hence (3.3) is fulfilled.

The analogue of lemma 2.9 for two intersecting null hypersurfaces requires, in addition to (1.13), the vanishing of

Equation (3.8)

Both (1.13) and (3.8) together imply vanishing initial data for the analogue of the ODE (2.73) in the current setting. Equation (3.8) follows from (1.7), the corresponding equation on N2, and (1.10). Thus (3.5) is fulfilled.

A straigthforward adaptation of lemma 2.11 to the current setting shows that, say on N2, $g^{AB}S_{AB2}-\frac{1}{2}\tau g^{12}A_{22}$ vanishes, supposing that it vanishes on S. Using lemma 2.3 (i) we find

because of (1.6)–(1.12). Hence also (3.6) is fulfilled. □

Remark 3.2. While τ has no zeros near the tip of a light-cone, for two transversally intersecting null hypersurfaces the expansion τ may vanish even near the intersection. In that case the trace of (3.3) on, say, N1 will fail to provide an algebraic equation for $\overline{X}{}^2$. Also, corollary 2.8 cannot be applied to deduce the vanishing of $\overline{S}{}_{221}$, equivalently, the validity of (3.4), in the regions where τ vanishes. Instead one can use the second-order ODE (3.4) to find a candidate for $\overline{X}{}^2$, and then lemma 2.7 guarantees that the trace of the left-hand side of (3.3) vanishes when gABAAB|S = 0 = ∂2(gABAAB)|S.

3.3. Bifurcate horizons

A key notion to the understanding of the geometry of stationary black holes is that of a bifurcation surface. This is a smooth submanifold S of co-dimension two on which a Killing vector X vanishes, with S forming a transverse intersection of two smooth null hypersurfaces so that X is tangent to the generators of each. In our context this would correspond to a KID which vanishes on S, and is tangent to the null generators of the two characteristic hypersurfaces emanating normally from S. In coordinates adapted to one of the null hypersurfaces, so that the hypersurface is given by the equation x1 = 0, we have $\overline{X} = \overline{X}{}^ 2 \partial _2$, and $\overline{X}{}^ at = \overline{g}{}_{12} \overline{X}{}^ 2 {\rm d}x^ 1$. Then (2.84) holds, and therefore also (2.88)–(2.90) (which correspond to (3.2) and (3.3)). Equations (2.89)–(2.90) show that this is only possible if τ = σAB = 0, which implies that translations along the generators of the light-cone are isometries of the (n − 1)-dimensional metric $\overline{g}{}_{AB} {\rm d}x^ A {\rm d}x^ B$. Equivalently, N1 and N2 have vanishing null second fundamental forms, which provides a necessary condition for a bifurcate horizon.

Assuming vacuum (as everywhere else in this work), this condition turns out to be sufficient. Let ζA be the torsion one-form of S (see, e.g., [3], or (B.4) below). We can use theorem 2.1 to prove (compare [8, proposition B.1] in dimension 3 + 1 and [9, end of section 2] in higher dimensions):

Theorem 3.3. Within the setup of theorem 2.1, suppose that the null second fundamental forms of the hypersurfaces Na, a = 1, 2, vanish. Then:

  • (i)  
    There exists a Killing vector field X defined on D+(N1N2) which vanishes on S and is null on N1N2.
  • (ii)  
    Furthermore, any Killing vector $\hat{Y} = \hat{Y}^A\partial _A$ of the metric induced by g on S extends to a Killing vector X of g on D+(N1N2) if and only if the $\hat{Y}$-Lie derivative of the torsion one-form of S is exact.

Remarks 3.4. 

  • (1)  
    Killing vectors as above would exist to the past of S if the past-directed null hypersurfaces emanating from S also had vanishing second null fundamental forms. However, this does not need to be the case, a vacuum example is provided by suitable Robinson– Trautman spacetimes.
  • (2)  
    Concerning point 2. of the theorem, when the $\hat{Y}$-Lie derivative of ζ is merely closed the argument of the proof below provides one-parameter families of Killing vectors defined on domains of dependence $D^+({\mathscr O})$ of simply connected subsets ${\mathscr O}$ of S. It would be of interest to find out whether or not the resulting locally defined Killing vectors can be patched together to a global one when S is not simply connected.

Proof. 

  • (1)  
    In coordinates adapted to the null hypersurfaces the condition that a Killing vector X is tangent to the generators is equivalent to
    Equation (3.9)
    Equation (3.10)
    For simplicity we assume that the generators of the two null hypersurfaces are affinely parameterized, i.e. $\kappa _{N_1}=\kappa _{N_2}=0$. By hypothesis we have
    Equation (3.11)
    The KID equations (1.6)–(1.13) for the candidate field $\overline{Y}{}$ reduce to
    Equation (3.12)
    Equation (3.13)
    Equation (3.14)
    Equation (3.15)
    Since $\overline{Y}{}_{\mu }|_S=0$ we need non-trivial initial data $\partial _2\overline{Y}{}_1|_{S}$ and $\partial _1\overline{Y}{}_2|_{S}$ for the ODEs (3.12) and (3.13) for $\partial _2\overline{Y}{}_1|_{N_1}$ and $\partial _1\overline{Y}{}_2|_{N_2}$, respectively.Using the formulae in [2, appendix A], (3.12)–(3.15) can be rewritten as
    Equation (3.16)
    Equation (3.17)
    Equation (3.18)
    Equation (3.19)
    Hence there remains the freedom to prescribe a constant c ≠ 0 for $\frac{1}{2}(\partial _1 \overline{Y}{}^1- \partial _2 \overline{Y}{}^2)|_S$. (The constant c reflects the freedom of scaling the Killing vector field by a constant, and is related to the surface gravity of the horizon; we will return to this shortly.) By (3.18) one needs to choose $\partial _1\overline{Y}{}^1|_S=c$ and $\partial _2\overline{Y}{}^2|_S=-c$. Together with $\overline{Y}{}^1|_S=\overline{Y}{}^2|_S=0$ the functions $\overline{Y}{}^2|_{N_1}$ and $\overline{Y}{}^1|_{N_2}$ are then determined by (3.16) and (3.17), and existence of the desired Killing vector follows from theorem 2.1.
  • (2)  
    By assumption we have, using obvious notation,
    Equation (3.20)

Now, the flow of a spacetime Killing vector field which is tangent to S preserves S. This implies that the bundle of null vectors normal to S is invariant under the flow. Equivalently, the image by the flow of a null geodesic normal to S will be a one-parameter family of null geodesics normal to S. This is possible only if the Killing vector field is tangent to both N1 and N2. It thus suffices to consider candidate Killing vector fields $\overline{Y}{}$ which satisfy, in our adapted coordinates,

Equation (3.21)

To continue, we need a simple form of the KID equations (1.6)–(1.13), assuming (3.20) and (3.21), and supposing again that the generators of the two null hypersurfaces are affinely parameterized, i.e. $\kappa _{N_1}=\kappa _{N_2}=0$. Using the notation $\hat{Y} \equiv Y^A|_S\partial _A$ we find:

Equation (3.22)

Equation (3.23)

Equation (3.24)

Equation (3.25)

Equation (3.26)

Equation (3.27)

(The fields g12|S and

Equation (3.28)

are part of the free initial data on S [11]; compare [5].) The first-order equations above are straightforward; some details of the derivation of the remaining equations above will be given shortly. Before passing to that, we observe that (3.22)–(3.23) and (3.26), together with (3.20), are equivalent to the requirement that $\hat{Y}|_S$ is a Killing vector field of (S, gAB|S), and that $\overline{Y}{}^A = \hat{Y}^A$ on N1N2, i.e. that $\overline{Y}{}^A$ is independent of the coordinates x1 and x2. Supposing further that ${\mathscr L}_{ \hat{Y}} \zeta _A$ is exact, the remaining equations (3.24), (3.25) and (3.27) can be used to determine $\overline{Y}{}^1$ and $\overline{Y}{}^2$ on N1N2. (As such, on each connected component of S the difference $(\partial _1\overline{Y}{}^1 - \partial _2\overline{Y}{}^2)|_S$ is determined up to an additive constant by (3.27), which reflects the freedom of adding a Killing vector field which vanishes on S and is tangent to the null geodesics generating both initial surfaces.) The existence of a Killing vector field X on the spacetime which coincides with $\hat{Y}$ on S follows now from theorem 1.2.

In appendix B it is shown that the KID equations (3.22)–(3.27) are invariant under affine reparameterizations of generators. We also show there that the freedom to choose an affine parameter on N1 and N2 can be employed to prescribe g12 on S, and also to add arbitrary gradients to ζ. In particular exactness of ζ, and thereby solvability of the KID equations, is independent of the gauge, as one should expect.

Let us pass to some details of the derivation of the second-order equations above. The calculation uses extensively

similarly on N2. First, let us compute $R_{122}{}^A|_{N_1}$ and $R_{211}{}^A|_{N_2}$, as needed to evaluate (1.9). Making use of [2, appendix A], we find

and thus

where, using the notation in [5], $\xi ^{N_1}_A=-2\Gamma ^2_{2A}|_{N_1}$ and $\xi ^{N_2}_A=-2\Gamma ^1_{1A}|_{N_2}$. Now for $\chi ^{N_1}_{AB}=\chi ^{N_2}_{AB}=0$ the vacuum characteristic constraint equations [2] imply $\partial _2\xi ^{N_1}_A=0$ and $\partial _1\xi ^{N_2}_A=0$, hence

Further simple calculations lead to (3.24).

Next, on S, we consider the KID equation

Equation (3.29)

Again on S it holds that

Inserting into (3.29) gives the following equation on S, in coordinates adapted to N2:

Equation (3.30)

The analogous formula in coordinates adapted to N1 reads

Equation (3.31)

Subtracting we obtain (3.27).

From the discussion so far it should be clear that the conditions are necessary. This concludes the proof. □

As an example, suppose that $\hat{Y}$ is a Killing vector on S and that the torsion one-form is invariant under the flow of $\hat{Y}$. It follows from the equations above that we can reparameterize the initial data surfaces so that g12 = 1 on S, with the torsion remaining invariant in the new gauge. Then $\overline{Y}{}^1=\overline{Y}{}^2=0$ and $\overline{Y}{}^A = \hat{Y}^A$ provides a solution of the KID equations on N1N2.

It is of interest to relate the constant c, arising in the paragraph following (3.19), to the surface gravity (which we denote by $\kappa _{\mathscr H}$ here); this will also prove in which sense the seemingly coordinate-dependent derivatives ∂1X1|S = −∂2X2|S are in fact geometric invariants. In the process we recover the well-known fact, that surface gravity is constant on bifurcate horizons. We have

Equation (3.32)

On, say, N1 we have due to (3.9) and (3.16)–(3.19)

Equation (3.33)

while $\nabla _1X^1|_{N_1}$ can be computed from (2.10),

where we used (3.9), the vanishing of χAB, and ∇1X1|S = c. Hence

Equation (3.34)

Acknowledgment

This work was supported in part by the Austrian Science Fund (FWF): P 24170-N16.

Appendix A.: Fuchsian ODEs

Since it appears difficult to find an adequate reference, we describe here the main property of first-order Fuchsian ODEs used in our work.

Consider a first order system of equations for a set of fields ϕ = (ϕI), I = 1, ..., N, of the form

Equation (A.1)

for some smooth map F = (FI) with F(0, 0) = 0, ∂ϕF(r, 0) = 0, where A(r) is a smooth map with values in N × N matrices. For our purposes it is sufficient to consider the case where

where  Id is the N × N identity matrix. It holds that the only solution of (A.1) such that limr → 0r−λϕ(r) = 0 is ϕ(r) = 0 for all r.

Appendix B.: Gauge-dependence of the torsion one-form

In this appendix we consider the question of gauge-independence in point (ii) of theorem 3.3. Indeed, even within the gauge conditions imposed so far, that x1 and x2 are affine parameters on the relevant characteristic surfaces, there is some gauge freedom left concerning the gravitational initial data. The point is that we can rescale the affine parameters x2 on N1 and x1 on N2,

Equation (B.1)

with some functions f± defined on S. Under (B.1), the metric on N1 becomes

Equation (B.2)

with a similar formula on N2. This leads to

Equation (B.3)

as well as, using [5, equation (2.12)],

Equation (B.4)

Letting $\check{x}^A = x^ A$, in the new coordinates the Killing vector becomes

Equation (B.5)

Invariance of (3.22)–(3.24) and (3.26) is clear. One can further check invariance of (3.25) (recall that $\hat{Y} \equiv Y^A\partial _A|_S$):

Equation (B.6)

As such, on S the first two-terms in (3.27) transform as

Equation (B.7)

Equation (B.4) can be rewritten as

Equation (B.8)

Thus

which shows that the one-form

Equation (B.9)

is invariant under changes of the affine parameters, as desired.

We end this paper by deriving the behavior of $\zeta _A\equiv \frac{1}{2}(\Gamma ^1_{1A}-\Gamma ^2_{2A})|_S$ under arbitrary coordinate transformations which preserve the adapted null coordinates conditions,

Equation (B.10)

We set

Then

i.e. (B.8) holds under coordinate transformations of the form (B.10).

If we assume S to be compact, there is a natural way to fix the gauge: according to the Hodge decomposition theorem ζ can be uniquely written as the sum of an exact one-form, a dual exact one-form and a harmonic one-form. The considerations above show that the first term has a pure gauge character and can be transformed away, while the remaining part has an intrinsic meaning. In particular, if ζ is exact, the remaining gauge freedom can be employed to transform it to zero.

Footnotes

  • Given a smooth vector field Yμ defined in a spacetime neighborhood of a hypersurface {f = 0} we will write $\overline{Y}{}^ \mu = Y^\mu |_{f=0}$, but at this point of the discussion $\overline{Y}{}^\mu$ is simply a vector field defined along the surface {f = 0}, it being irrelevant whether or not $\overline{Y}{}^ \mu$ arises by restriction of a smooth spacetime vector field. On the other hand, that last question will become a central issue in the proof of theorem 2.5.

  • We use the following conventions on indices: Greek indices are for spacetime tensors and coordinates, small Latin letters shall be used for tensors and coordinates on the light-cone or the characteristic surfaces, and capital Latin letters for tensors or coordinates in the hypersurfaces of spacetime co-dimension two foliating the characteristic surfaces.

  • Throughout we shall make extensively use of the formulae for the Christoffel symbols in adapted null coordinates computed in [2, appendix A]. Apart from the vanishing of the $\overline{\Gamma }{}^0_{i1}$'s the expressions for $\overline{\Gamma }{}^0_{01}$, $\overline{\Gamma }{}^1_{11}=\kappa$, $\overline{\Gamma }{}^A_{1B}$ and $\overline{\Gamma }{}^0_{AB}$ will be often used.

Please wait… references are loading.
10.1088/0264-9381/30/23/235036